Novel optical soliton molecules formed in a fiber laser with near-zero net cavity dispersion

Soliton molecules (SMs) are stable bound states between solitons. SMs in fiber lasers are intensively investigated and embody analogies with matter molecules. Recent experimental studies on SMs formed by bright solitons, including soliton-pair, soliton-triplet or even soliton-quartet molecules, are intensive. However, study on soliton-binding states between bright and dark solitons is limited. In this work, the formation of such novel SMs in a fiber laser with near-zero group velocity dispersion (ZGVD) is reported. Physically, these SMs are formed because of the incoherent cross-phase modulation of light and constitute a new form of SMs that are conceptually analog to the multi-atom molecules in chemistry. Our research results could assist the understanding of the dynamics of large SM complexes. These findings may also motivate potential applications in large-capacity transmission and all-optical information storage.


Section 1: A schematic of the fiber ring cavity.
Experimentally, to separately observe the two orthogonal polarization components of the light field, the laser output is first sent to a fiber pigtailed polarization beam splitter and then monitored with a high-speed detection system consisting of two 40GHz photodetectors (Newport, Model 1014) and a 33GHz bandwidth real-time oscilloscope (Agilent Technologies, DSA-93204 A). An extra polarization controller (PC2) is inserted between the laser output and the beam splitter to balance the linear polarization change induced by the lead fibers. Finally, an optical spectrum analyzer (Yokogawa, AQ6375) is used to monitor the optical spectrum of the laser emission.
To make the simulation results directly comparable with the experimental observations, our simulations were conducted based on the real experimental fiber laser configuration as shown in Fig. S1. We adopt a pulse tracing technique to simulate the laser operation 1. Briefly, when a light pulse circulates inside the cavity, the local fiber group velocity dispersion and birefringence varies with the fiber used. At different positions of the cavity, the pulse may have slightly different pulse shapes and energies. We note that as our cavity length is much shorter than the pulse dispersion and nonlinearity length, the dispersion-managed features are not dominant. Numerically, we have output the light pulse at different positions of the ring cavity and verified that the pulse properties are mainly determined by the average cavity parameters rather than that of the single segment of the fiber ring laser 2 .
Section 2: Stability of the trapped dark-bright vector solitons.  Fig. 6, when the wavelength difference between the dark and bright vector solitons becomes small so that their group velocity mismatch also becomes small, eventually the vector dark solitons will be trapped by the vector bright solitons and they will move as an entity inside the cavity, as shown in Fig. S2a. Fig. S2b shows the same vector dark-bright soliton pairs extracted at t = 60 s. It shows that this trapping state is very stable. From the experimental result it is also to see, if once formed, the temporal offset between the dark and bright solitons nearly remains unchanged, suggesting a stable bond between them.

Section 3: Investigation on indirect soliton interactions.
In the article, we have focused our studies on the short-range soliton interactions between dark and bright solitons. In particular, we showed that three types of "polyatomic soliton molecules" could be formed as a result of direct soliton interactions between the dark and bright solitons, either along orthogonally polarized axes (XPM coefficient σ=2/3) or along the same polarization axis but with different central wavelengths (XPM coefficient σ=2). In parallel with the short-range soliton interactions, in the case of moderate soliton separations, the solitons could also form weaker bonds caused by long-range interactions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] . In the following section, we will provide some experimental evidence to show the existence of the long-range soliton interactions under certain laser operation conditions and their influences on the observed soliton dynamics.
We start from a typical bright soliton emission state as shown in Fig  have showed that under strong coupling between dark and bright solitons, a type of 2+2 PSMs could even be formed as presented in Fig. 7. It is to note that the existence of unstable CW or strong DWs will unavoidably influence the dark-bright soliton dynamics through the indirect soliton interactions (long-range interactions). Therefore, similar to the formation of the state shown in Fig. S3c, random irregularly distributed dark-bright soliton molecules as shown in Fig. S4 could be formed.  Where 0 is the small signal gain coefficient and is the saturation energy. We adopted a pulse tracing technique to simulate the laser operation. Briefly, we start with a certain initial light condition, and let the light circulate in the cavity. In different cavity fibers, we used the parameters of the individual fiber for the calculation. When the light meets the cavity output port, 10% of the light intensity is deducted from the light fields. After one cavity roundtrip, the light is then reinjected into the cavity as the input for the next round calculation. We used the standard split-step method to solve the coupled extended CGLEs (4.1). The numerical calculations were made on a 400 ps window and the periodic boundary condition was used. As mentioned, we also simulated the interactions between dark and bright solitons polarized along the same polarization axis by setting the XPM coupling coefficient in the E.q. (4.1) to 2, Noteworthy mentioning that although our current simulations are based on the CGLEs, under appropriate conditions, for example, if the gain bandwidth is larger than the soliton spectral bandwidth and the laser gain is balanced by the cavity losses, the CGLEs could be mathematically reduced to CNLSE 19 .   In the manuscript, we have numerically studied the soliton interactions between dark and bright solitons along the orthogonally polarized axis. Numerically, we also studied interactions between dark and bright solitons polarized along the same polarization direction. In this case the XPM coupling coefficient σ in Eq. (1) is changed to 2. We start the simulation with two weak bright and dark pulses. Each of the bright pulses has the ℎ( ( + )) form, and each of the dark pulses has the ℎ( ) form. Here, dt represents an initial temporal separation between the dark and bright pulses. The bright solitons propagate in the net anomalous dispersion regime, while the dark solitons propagate in the net normal dispersion regime.

Section 5: Simulation results on coexistence of dark and bright solitons.
Initially, they propagate in opposite directions in the cavity, as shown in Fig. S6. Under large group velocity mismatches, they travel independently in the cavity. The results coincide well with the experimental results, as shown in Fig. 6. We also numerically verified that if the coupling force between them is strong, the dark and bright solitons could be coupled with each other. Consequently, they could be trapped and co-propagate in the cavity, as shown in Supplementary Fig. S7.
Tab. S1. Parameters used in the Fig. S6.  Fig. S6, we decreased the walk off between the dark and bright pulses by setting the beat length to be = 5 , consequently, the group velocity mismatch reduced to = 0.0005 −1 . With the smaller group velocity mismatch, the coupling between the dark and bright pulses become stronger, as a result, each of the dark soliton is captured by a bright soliton, they form a typical "1+1 soliton molecule" consisting of a scalar dark and a scalar bright soliton. Numerically we found that if δ is set as a negative value, corresponding to that the dark pulse travels faster than the bright pulse, the formed bound solitons would be that the dark soliton leads the bright soliton.
This result could be used to explain why two forms of (2+1) PSMs could be observed experimentally. Specifically, if the scalar dark soliton travels faster than the ODBS, the PSM will have the form as shown in Fig. 5a, while in the opposite condition, the PSM will feature those as shown in Fig. 5c.